Question 1

Solve the exponential equation. Express the solution set in terms of natural logarithms.
-$$5 ^ { x + 6 } = 2$$

Accepted Answer

A)  $\left\{ \frac { \ln 2 } { \ln 5 } - 6 \right\}$ 
B)  $\left\{ \frac { \ln 5 } { \ln 2 } + 6 \right\}$ 
C)  $\left\{ \frac { \ln 5 } { \ln 2 } + \ln 6 \right\}$ 
D)  $\{ \ln 2 - \ln 5 - \ln 6 \}$ 
A)  $\left\{ \frac { \ln 2 } { \ln 5 } - 6 \right\}$ 
B)  $\left\{ \frac { \ln 5 } { \ln 2 } + 6 \right\}$ 
C)  $\left\{ \frac { \ln 5 } { \ln 2 } + \ln 6 \right\}$ 
D)  $\{ \ln 2 - \ln 5 - \ln 6 \}$

Question 2

Use the Product Rule
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
-$\log _ { 2 } ( 4 x )$

Accepted Answer

A)  $2 + \log _ { 2 } x$ 
B)  $4 + \log _ { 2 } x$ 
C)  $2 x$ 
D)  $2 \log _ { 2 } x$ 
A)  $2 + \log _ { 2 } x$ 
B)  $4 + \log _ { 2 } x$ 
C)  $2 x$ 
D)  $2 \log _ { 2 } x$

Question 3

Use the One-to-One Property of Logarithms to Solve Logarithmic Equations
Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic
expressions. Give the exact answer.
-$\log 4 x = \log 5 + \log ( x - 1 )$

Accepted Answer

A)  $\{ 5 \}$ 
B)  $\left\{ \frac { 4 } { 3 } \right\}$ 
C)  $\{ - 5 \}$ 
D)  $\left\{ - \frac { 5 } { 9 } \right\}$ 
A)  $\{ 5 \}$ 
B)  $\left\{ \frac { 4 } { 3 } \right\}$ 
C)  $\{ - 5 \}$ 
D)  $\left\{ - \frac { 5 } { 9 } \right\}$

Question 4

Solve the exponential equation. Express the solution set in terms of natural logarithms.
-$$4 ^ { x + 4 } = 5 ^ { 2 x + 5 }$$

Accepted Answer

A)  $\left\{ \frac { 5 \ln 5 - 4 \ln 4 } { \ln 4 - 2 \ln 5 } \right\}$ 
B)  $\left\{ \ln \left[ \frac { 5 ^ { 5 } } { 4 ^ { 4 } } - \frac { 4 } { 5 ^ { 2 } } \right] \right\}$ 
C)  $\{ \ln 5 - \ln 4 \}$ 
D)  $\{ 7 \ln 5 - 5 \ln 4 \}$ 
A)  $\left\{ \frac { 5 \ln 5 - 4 \ln 4 } { \ln 4 - 2 \ln 5 } \right\}$ 
B)  $\left\{ \ln \left[ \frac { 5 ^ { 5 } } { 4 ^ { 4 } } - \frac { 4 } { 5 ^ { 2 } } \right] \right\}$ 
C)  $\{ \ln 5 - \ln 4 \}$ 
D)  $\{ 7 \ln 5 - 5 \ln 4 \}$

Question 5

Use the Product Rule
Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
-$\log ( 10,000 x )$

Accepted Answer

A)  $4 + \log x$ 
B)  $40 + \log x$ 
C)  $4 x$ 
D)  $4 \log x$ 
A)  $4 + \log x$ 
B)  $40 + \log x$ 
C)  $4 x$ 
D)  $4 \log x$

Question 6

Use Compound Interest Formulas
Use the compound interest formulas A $$A = P \left( 1 + \frac { r } { n } \right) \text { and } A = P e ^ { r t } \text { to solve }$$
-Suppose that you have $10,000 to invest. Which investment yields the greater return over 6 years: 5.4% compounded monthly or 5.5% compounded quarterly?

Accepted Answer

A) $10,000 invested at 5.5% compounded quarterly over 6 years yields the greater return. 
B) $10,000 invested at 5.4% compounded monthly over 6 years yields the greater return. 
C) Both investment plans yield the same return. 
A) $10,000 invested at 5.5% compounded quarterly over 6 years yields the greater return. 
B) $10,000 invested at 5.4% compounded monthly over 6 years yields the greater return. 
C) Both investment plans yield the same return.

Question 7

Graph the function.
-Use the graph of $f ( x ) = 2 ^ { x }$ to obtain the graph of $g ( x ) = 2 ^ { x } + 2$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 8

Write the equation in its equivalent exponential form.
-$$\log _ { 6 } 216 = x$$

Accepted Answer

A)  $6 ^ { x } = 216$ 
B)  $x ^ { 6 } = 216$ 
C)  $216 ^ { x } = 6$ 
D)  $216 ^ { 6 } = x$ 
A)  $6 ^ { x } = 216$ 
B)  $x ^ { 6 } = 216$ 
C)  $216 ^ { x } = 6$ 
D)  $216 ^ { 6 } = x$

Question 9

Solve the problem.
-The $\mathrm { pH }$ of a solution ranges from 0 to 14. An acid has a pH less than 7. Pure water is neutral and has a pH of 7. The pH of a solution is given by $\mathrm { pH } = - \log \mathrm { x }$ where $\mathrm { x }$ represents the concentration of the hydrogen ions in the solution in moles per liter. Find the $\mathrm { pH }$ if the hydrogen ion concentration is $1 \times 10 ^ { - 3 }$.

Accepted Answer

A)  3 
B)  11 
C)  $- 3$ 
D)  $- 11$ 
A)  3 
B)  11 
C)  $- 3$ 
D)  $- 11$

Question 10

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places
-$\log _ { \pi } 18$

Accepted Answer

A)  $2.5249$ 
B)  $0.3960$ 
C)  $1.7524$ 
D)  $0.7581$ 
A)  $2.5249$ 
B)  $0.3960$ 
C)  $1.7524$ 
D)  $0.7581$

Question 11

Solve the equation by expressing each side as a power of the same base and then equating exponents.
-$$32 ^ { x } = \frac { 1 } { \sqrt { 2 } }$$

Accepted Answer

A)  $\left\{ - \frac { 1 } { 10 } \right\}$ 
B)  $\left\{ \frac { 1 } { 10 } \right\}$ 
C)  $\left\{ - \frac { 1 } { 5 } \right\}$ 
D)  $\{ - 5 \}$ 
A)  $\left\{ - \frac { 1 } { 10 } \right\}$ 
B)  $\left\{ \frac { 1 } { 10 } \right\}$ 
C)  $\left\{ - \frac { 1 } { 5 } \right\}$ 
D)  $\{ - 5 \}$

Question 12

Write the equation in its equivalent exponential form.
-$\log _ { 5 } x = 3$

Accepted Answer

A)  $5 ^ { 3 } = x$ 
B)  $35 = x$ 
C)  $5 x = 3$ 
D)  $x ^ { 3 } = 5$ 
A)  $5 ^ { 3 } = x$ 
B)  $35 = x$ 
C)  $5 x = 3$ 
D)  $x ^ { 3 } = 5$

Question 13

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm
whose coefficient is 1. Where possible, evaluate logarithmic expressions.
-$2 \ln a - 9 \ln b$

Accepted Answer

A)  $\ln \frac { a ^ { 2 } } { b ^ { 9 } }$ 
B)  $\frac { \ln a ^ { 2 } } { \ln b ^ { 9 } }$ 
C)  $\ln \left( \frac { \mathrm { a } } { \mathrm { b } } \right) ^ { 11 }$ 
D)  $\ln \frac { 2 a } { 9 b }$ 
A)  $\ln \frac { a ^ { 2 } } { b ^ { 9 } }$ 
B)  $\frac { \ln a ^ { 2 } } { \ln b ^ { 9 } }$ 
C)  $\ln \left( \frac { \mathrm { a } } { \mathrm { b } } \right) ^ { 11 }$ 
D)  $\ln \frac { 2 a } { 9 b }$

Question 14

Write the equation in its equivalent logarithmic form.
-$$\sqrt [ 3 ] { 8 } = 2$$

Accepted Answer

A)  $\log _ { 8 } 2 = \frac { 1 } { 3 }$ 
B)  $\log _ { 2 } 8 = \frac { 1 } { 3 }$ 
C)  $\log _ { 8 } 3 = \frac { 1 } { 2 }$ 
D)  $\log _ { 2 } 8 = 3$ 
A)  $\log _ { 8 } 2 = \frac { 1 } { 3 }$ 
B)  $\log _ { 2 } 8 = \frac { 1 } { 3 }$ 
C)  $\log _ { 8 } 3 = \frac { 1 } { 2 }$ 
D)  $\log _ { 2 } 8 = 3$

Question 15

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm
whose coefficient is 1. Where possible, evaluate logarithmic expressions.
-$$\frac { 1 } { 5 } \left[ 5 \ln ( x + 1 ) - \ln x - \ln \left( x ^ { 2 } - 1 \right) \right]$$

Accepted Answer

A)  $\ln \sqrt [ 5 ] { \frac { ( x + 1 ) ^ { 5 } } { x \left( x ^ { 2 } - 1 \right) } }$ 
B)  $\ln \sqrt [ 5 ] { \frac { ( x + 1 ) ^ { 5 } \left( x ^ { 2 } - 1 \right) } { x } }$ 
C)  $\ln \sqrt [ 5 ] { \frac { 5 ( x + 1 ) } { x \left( x ^ { 2 } - 1 \right) } }$ 
D)  $\ln \sqrt [ 5 ] { \frac { x ( x + 1 ) ^ { 5 } } { \left( x ^ { 2 } - 1 \right) } }$ 
A)  $\ln \sqrt [ 5 ] { \frac { ( x + 1 ) ^ { 5 } } { x \left( x ^ { 2 } - 1 \right) } }$ 
B)  $\ln \sqrt [ 5 ] { \frac { ( x + 1 ) ^ { 5 } \left( x ^ { 2 } - 1 \right) } { x } }$ 
C)  $\ln \sqrt [ 5 ] { \frac { 5 ( x + 1 ) } { x \left( x ^ { 2 } - 1 \right) } }$ 
D)  $\ln \sqrt [ 5 ] { \frac { x ( x + 1 ) ^ { 5 } } { \left( x ^ { 2 } - 1 \right) } }$

Question 16

Graph the function.
-Use the graph of $f ( x ) = 4 ^ { x }$ to obtain the graph of $g ( x ) = - 4 ^ { x }$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 17

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places
-$\log _ { 20 } 66.6$

Accepted Answer

A)  $1.4016$ 
B)  $0.7135$ 
C)  $3.1245$ 
D)  $0.5224$ 
A)  $1.4016$ 
B)  $0.7135$ 
C)  $3.1245$ 
D)  $0.5224$

Question 18

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate
logarithmic expressions without using a calculator.
-$\ln \sqrt { \frac { x } { y } }$

Accepted Answer

A)  $\frac { 1 } { 2 } \ln x - \frac { 1 } { 2 } \ln y$ 
B)  $\ln \sqrt { x } - \ln \sqrt { y }$ 
C)  $\frac { 1 } { 2 } \ln \frac { x } { y }$ 
D)  $\frac { 1 } { 2 } \ln x - \ln y$ 
A)  $\frac { 1 } { 2 } \ln x - \frac { 1 } { 2 } \ln y$ 
B)  $\ln \sqrt { x } - \ln \sqrt { y }$ 
C)  $\frac { 1 } { 2 } \ln \frac { x } { y }$ 
D)  $\frac { 1 } { 2 } \ln x - \ln y$

Question 19

Find the domain of the logarithmic function.
-$$f ( x ) = \log _ { 5 } ( x - 5 )$$

Accepted Answer

A)  $( 5 , \infty )$ 
B)  $( - \infty , 5 )$ or $( 5 , \infty )$ 
C)  $( - 5 , \infty )$ 
D)  $( - \infty , 0 )$ or $( 0 , \infty )$ 
A)  $( 5 , \infty )$ 
B)  $( - \infty , 5 )$ or $( 5 , \infty )$ 
C)  $( - 5 , \infty )$ 
D)  $( - \infty , 0 )$ or $( 0 , \infty )$

Question 20

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm
whose coefficient is 1. Where possible, evaluate logarithmic expressions.
-$\log x + \log 2$

Accepted Answer

A)  $\log 2 x$ 
B)  $( \log x ) ( \log 2 )$ 
C)  $\log \frac { x } { 2 }$ 
D)  $\log ( x + 2 )$ 
A)  $\log 2 x$ 
B)  $( \log x ) ( \log 2 )$ 
C)  $\log \frac { x } { 2 }$ 
D)  $\log ( x + 2 )$

Solve the exponential equation. Express the solution set in terms of natural logarithms. - $5 ^ { x + 6 } = 2$

Use the Product Rule Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\log _ { 2 } ( 4 x )$

Use the One-to-One Property of Logarithms to Solve Logarithmic Equations Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic expressions. Give the exact answer. - $\log 4 x = \log 5 + \log ( x - 1 )$

Solve the exponential equation. Express the solution set in terms of natural logarithms. - $4 ^ { x + 4 } = 5 ^ { 2 x + 5 }$

Use the Product Rule Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\log ( 10,000 x )$

Graph the function. -Use the graph of $f ( x ) = 2 ^ { x }$ to obtain the graph of $g ( x ) = 2 ^ { x } + 2$ $Graph the function. -Use the graph of f ( x ) = 2 ^ { x } to obtain the graph of g ( x ) = 2 ^ { x } + 2$

Write the equation in its equivalent exponential form. - $\log _ { 6 } 216 = x$

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places - $\log _ { \pi } 18$

Solve the equation by expressing each side as a power of the same base and then equating exponents. - $32 ^ { x } = \frac { 1 } { \sqrt { 2 } }$

Write the equation in its equivalent exponential form. - $\log _ { 5 } x = 3$

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions. - $2 \ln a - 9 \ln b$

Write the equation in its equivalent logarithmic form. - $\sqrt [ 3 ] { 8 } = 2$

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions. - $\frac { 1 } { 5 } \left[ 5 \ln ( x + 1 ) - \ln x - \ln \left( x ^ { 2 } - 1 \right) \right]$

Graph the function. -Use the graph of $f ( x ) = 4 ^ { x }$ to obtain the graph of $g ( x ) = - 4 ^ { x }$ $Graph the function. -Use the graph of f ( x ) = 4 ^ { x } to obtain the graph of g ( x ) = - 4 ^ { x }$

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places - $\log _ { 20 } 66.6$

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - $\ln \sqrt { \frac { x } { y } }$

Find the domain of the logarithmic function. - $f ( x ) = \log _ { 5 } ( x - 5 )$

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions. - $\log x + \log 2$

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Exam 4: Exponential and Logarithmic Functions

Solve the exponential equation. Express the solution set in terms of natural logarithms. - 5x+6=25 ^ { x + 6 } = 25x+6=2

Use the Product Rule Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - log⁡2(4x)\log _ { 2 } ( 4 x )log2​(4x)

Solve the exponential equation. Express the solution set in terms of natural logarithms. - 4x+4=52x+54 ^ { x + 4 } = 5 ^ { 2 x + 5 }4x+4=52x+5

Use the Product Rule Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - log⁡(10,000x)\log ( 10,000 x )log(10,000x)

Graph the function. -Use the graph of f(x)=2xf ( x ) = 2 ^ { x }f(x)=2x to obtain the graph of g(x)=2x+2g ( x ) = 2 ^ { x } + 2g(x)=2x+2

Write the equation in its equivalent exponential form. - log⁡6216=x\log _ { 6 } 216 = xlog6​216=x

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places - log⁡π18\log _ { \pi } 18logπ​18

Solve the equation by expressing each side as a power of the same base and then equating exponents. - 32x=1232 ^ { x } = \frac { 1 } { \sqrt { 2 } }32x=2​1​

Write the equation in its equivalent exponential form. - log⁡5x=3\log _ { 5 } x = 3log5​x=3

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions. - 2ln⁡a−9ln⁡b2 \ln a - 9 \ln b2lna−9lnb

Write the equation in its equivalent logarithmic form. - 83=2\sqrt [ 3 ] { 8 } = 238​=2

Graph the function. -Use the graph of f(x)=4xf ( x ) = 4 ^ { x }f(x)=4x to obtain the graph of g(x)=−4xg ( x ) = - 4 ^ { x }g(x)=−4x

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places - log⁡2066.6\log _ { 20 } 66.6log20​66.6

Use properties of logarithms to expand the logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. - ln⁡xy\ln \sqrt { \frac { x } { y } }lnyx​​

Find the domain of the logarithmic function. - f(x)=log⁡5(x−5)f ( x ) = \log _ { 5 } ( x - 5 )f(x)=log5​(x−5)

Use properties of logarithms to condense the logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions. - log⁡x+log⁡2\log x + \log 2logx+log2